1. AP Physics B: Ch.10 - Elasticity and Simple Harmonic Motion Reading Assignment Cutnell and Johnson, Physics Chapter 10

2. Simple Harmonic motion/systems Mechanical systems use springs extensively Imagine a mechanical car engine – lots of moving parts – back and forth movement. Prototype for all vibrating systems is simplest possible case called Simple Harmonic Motion (SHM)

3. Simple Harmonic motion Important for understanding many disparate phenomena, e.g., vibration of mechanical structures (bridges, cars, buildings) Electrical radio receivers

4. Springs Obey Hooke’s law in both extension and compression, i.e., F = k x where k is spring constant [N/m] k is measure of the stiffness of the spring. In other words the restoring force of the spring depends on its strength and how far it is extended © John Wiley and Sons, 2004

5. Springs Newton’s Third Law => The restoring force of an ideal spring is F = -k x k is spring constant, x is displacement from unstrained length. If I exert a force to stretch a spring, the spring must exert an equal but opposite force on me: “restoring force”. Restoring force is always in opposite direction to displacement of the spring.

6. Springs Stretching an elastic material or spring, and then releasing leads to oscillatory motion. In absence of air resistance or friction we get “Simple Harmonic Motion” © John Wiley and Sons, 2004

7. Springs The motion of the spring in the diagram below is said to be simple harmonic motion. If a pen is attached to an object in simple harmonic motion then the output will be a sine wave. © John Wiley and Sons, 2004

8. Springs Simple harmonic motion is where the acceleration of the moving object is proportional to its displacement from its equilibrium position. It is the oscillation of an object about its equilibrium position The motion is periodic and can be described as that of a sine function (or equivalently a cosine function), with constant amplitude. It is characterised by its amplitude, its period and its phase.

9. Springs The motion is periodic and can be described as that of a sine function (or equivalently a cosine function), with constant amplitude. It is characterised by its amplitude, its period and its phase. © John Wiley and Sons, 2004

10. Springs Simple harmonic motion equation –  - is a constant (angular velocity) © John Wiley and Sons, 2004

11. Simple Pendulum Pivot mg  L l s Tension Restoring force acts to pull bob back towards vertical. mg sin  is the restoring force F= ma s so restoring force = ma s = mg sin  mg cos  mg sin 

12. Result: Compare with acceleration of mass on a spring. Time for one oscillation (T) Equations are of same form, and we can say for pendulum is equivalent to simple harmonic motion.

13. Elasticity Experiment m Vernier scale Rigid beam Metal wire mg Results

14. Elasticity Experiment - Results

15. Definitions Tensile stress =applied force per unit area = F/A [N/m 2 ] Tensile strain = extension per unit length =  L/ L o Experiments show that, up to the elastic limit, Tensile stress  Tensile strain (Hooke’s Law) i.e.,

16. Definitions Young’s Modulus = Stress/Stain

17. To measure Young’s Modulus for brass Measure: L 0 = 1.00 m A = 0.78 x 10 -6 m 2 slope = 1.43 x 10 -5 m/N => Y =

18. Typical Young’s modulus values

19. Damped harmonic Motion In reality, non-conservative forces always dissipate energy in the oscillating system. This is called “damping”.

22. Resonance Resonance is the condition where a driving force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. In the absence of damping, resonance occurs when the frequency of the driving force matches the natural frequency of oscillation of the object.

23. Pushing a swing at the correct time…/incorrect time…resonance

24. Summary Materials subject to elastic deformation obey Hooke’s Law Stretching an elastic material or spring, and then releasing leads to oscillatory motion. In absence of air resistance or friction we get “Simple Harmonic Motion”